2PA3 EXPERIMENT 3a
version 23 Sep 2003
HEAT CAPACITY RATIO FOR GASES (g)
OBJECTIVE: Measure the speed of sound in various gases, and determine the heat
capacity ratio, g, for nitrogen, carbon dioxide, helium and argon by both the
adiabatic expansion (Clement and Desormes) and speed of sound methods.
1.
INTRODUCTION:
By definition CV (or Cp) denotes the amount of heat energy which must be
absorbed by one mole of a gas at constant volume (or pressure) to raise the
temperature of the gas by one degree. The absorbed heat energy causes the
molecules to move faster (increase in translational energy), to rotate faster (increase
in rotational energy) and to vibrate faster (increase in vibrational energy). Thus the
knowledge of heat capacities plays a role in understanding the complexity of
gaseous molecules. Unfortunately the easiest method for determining the individual
heat capacities of gases is beyond the capability of an undergraduate laboratory. The
heat capacity ratio, g = Cp/CV, is just as useful in understanding the structure of
gaseous molecules and is more accessible experimentally.
2.
CLEMENT AND DISORMES METHOD FOR g
2.1 Theory:
Consider the following two stage process carried out on a gas:
Stage I is a reversible adiabatic expansion of the gas from V1 to V2 with cooling
from T1 to T2 since the gas is doing work of expansion and no compensating heat
transfer is allowed.
Thus:
dU = dq + dw
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Heat Capacity Ratio for Gases (g)
2
dq = 0 for an adiabatic transformation, dw = -PdV for a reversible expansion
 dU = - PdV
(1)
The total differential of U is given by
dU = (
U
U
)v dT + (
)T dV
T
V
(2)
If we assume the gas is ideal
 
dU
dV
=0
T
and
dU = (
U
)V dT = n CV dT
T
 n C V dT = - PdV
(3)
(4)
and again for an ideal gas PV = nRT so
n CV
dT
dV
= - nR
T
V
(5)
integrating we find
n CV ln
T2
V
=−n R ln 2
T1
V1
or since
T 2 P2 V 2
=
T 1 P1 V 1
for an ideal gas
(6)
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Heat Capacity Ratio for Gases (g)
CV l n
since, for an ideal gas
V
P2
V
= - ( CV + R) l n 2  C p ln 2
V1
P1
V1
CV + R = C p
3
(7)
(8)
therefore
l n ( P1 / P2 )
Cp
= g =
l n (V 2 / V 1)
CV
(9)
Stage II involves heat transfer to the gas at constant volume sufficient to restore the
temperature to T1. Thus, assuming ideal gas behaviour,
P3
T1
P1 V 1
=
=
P2
T2
P2 V 2
1
(10)
V2
P1
=
V1
P3
(11)
Thus
g =
2
l n( P 1 / P 2 )
l n ( P1 / P 3 )
(12)
2.2 Experimental Method:
In this experiment we take a quantity of gas in a large bottle (carboy) at
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4
pressure P1 (greater than atmospheric) and temperature T1 and remove the stopper
for an instant to allow the gas to expand adiabatically to pressure P2, (atmospheric),
cooling to T2 (See Fig. 1).
Figure 1 Clement and Désormes Method for Measuring g
The stopper is immediately replaced and the gas is allowed to warm up to the
original temperature, T1 at constant volume V2 and the final pressure P3 is measured.
From the measured values of P1, P2 and P3, the heat capacity ratio, g, can be
calculated.
Note that the quantity of gas under consideration is not the total initially in the bottle
but only the portion V1 below an imaginary boundary which subsequent to stage I
completely fills the bottle. The remainder is lost from the bottle and not accounted
for.
Figure 2. Clement & Désormes Apparatus
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2.3 Procedure:
Assemble the apparatus as shown in Fig. 2. Note that the working fluid in the openend manometer is dibutyl phthalate. All manometer readings must be converted to
equivalent readings in cm Hg and then added to the barometric pressure.
1) Open clamp B and the ball valve C. Set the gas regulator to between 5 and 10
psig gas pressure.
2) Slowly open the needle valve A and sweep out the carboy with the gas for 15
minutes. Reduce the gas flow by partially closing A.
3) Close the ball valve C and pressurise the carboy to near the maximum pressure
allowed by the manometer, taking care not to blow the manometric fluid out of the
glass tube.
4) Finally close B. Initially the pressure usually falls. This is due to cooling of the
gas which has warmed as a result of work done on it in the pressurising process
(bicycle pump effect). It may also fall due to a leak in the system. Allow the gas to
come to thermal equilibrium with the surroundings as indicated by constancy of
pressure.
5) Record the levels of dibutylphthalate in the manometer.
6) Expand the gas adiabatically by quickly opening and closing the ball valve C.
Allow the gas to warm up to room temperature and record the manometer readings.
7) Repeat the entire procedure two more times with the same gas, but omit the
flushing unless air has got into the carboy. Record the ambient barometric pressure
and temperature. Note: Do not open windows during the experiment.
temperature of the surroundings must be kept as constant as possible. (Why?)
The
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2.4 Calculations:
Calculate g for each run and determine the average value of g. Calculate g for
each gas from the Equipartition of Energy Principle and the number of degrees of
freedom. Identify the various degrees of freedom of each gas studied. Calculate the
order of magnitude of the temperature drop of gas at the moment of expansion
(assume an ideal gas).
SPEED OF SOUND METHOD FOR g (KUNDT'S TUBE)
3.1 Theory:
Only a brief outline of the theory underlying the measurement of the speed of
sound is given here. [For a more detailed discussion of the subject matter the student
is referred to more specialized texts (1) dealing with sound.] Sound-waves require
elastic media for their propagation, for a train of sound-waves consists essentially of
a sequence of elastic displacements of elements of the medium.
These
displacements take place in the direction of propagation so that sound-waves are
longitudinal in nature. The longitudinal waves are produced by a disturbance
applied to the elastic medium causing compressions and rarefactions. Fig. 3 shows
elements of an elastic medium at "rest". A stimulus when applied to the medium at
rest, causes the elements to vibrate about their mean positions and, because they do
not vibrate in phase, this causes pressure (hence density) differences in the medium
resulting in compressions and rarefactions along the wavefront, as denoted by the
letters C and R in Fig.3. These compressions and rarefactions are taking place with
such rapidity that the process for all intents and purposes is adiabatic; there is no
time for heat exchange between the vibrating elements and their surroundings, i.e. q
= 0. The adiabatic compressions and expansions of ideal gases are governed by
3.
PVg = constant
(c.f. PV = constant for an isothermal expansion of an ideal gas)
(13)
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Fig. 3 (a) elements of elastic medium at "rest". (b) a disturbance passing
through the medium causes compressions (C) and rarefactions (R).
The speed of sound in any medium depends on the elasticity of the medium and its
density.
It can be shown (1) that the speed of sound, c, in any medium is given by the
equation
c =
K /
(14)
where K is the bulk modulus of elasticity of the medium and D its density. For a gas
undergoing abiabatic pressure changes the bulk modulus of elasticity is simply (P,
where P is the (absolute) pressure of the gas. Thus
c =
gP / 
(15)
If an assumption be made that the gas behaves ideally, then PV = nRT also applies,
which upon rearrangement gives
P
RT
=

M
where M is the molar mass of the gas.
(16)
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Substitution of (3.16) into (3.15), gives
c =
gRT
M
(17)
g =
M c2
RT
(18)
or
If the distance between corresponding points on the wave front (e.g. maximum
compressions) is called the wave-length, l, and the rate at which such compressions
pass a given section is called the frequency, f, then the speed of sound is also given
by
c = fl
(19)
and
g =
M f 2 l2
RT
(20)
In this experiment, which is based on a modified Kundt's tube, a sound-wave of
known frequency is sent down a long glass tube (2) and is made to reflect from a
movable piston creating a standing wave with a series of maxima, called antinodes,
A, and minimum points in between, called nodes, N, Fig. 4.
Fig. 4 A standing wave with nodes, N, and antinodes, A, created by
reflection of a wave from the piston P.
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The wave appears to be standing which is a result of interference between two waves
of the same frequency travelling with the same speed in opposite directions. The
distance between nearest nodes (or anti-nodes) is equal to l/2. The successive nodes
(or anti-nodes) are in opposite phase, i.e. they differ in phase by 180 degrees (a fact
which we will take advantage of in this experiment), with maximum sound intensity
occurring at the nodes and minimum at the anti-nodes.
3.2 Procedure:
The apparatus is shown in Fig. 5
Fig. 5 Kundt's Tube
1) Move the piston furthest away from the speaker and let the gas sweep out air from
the Kundt's tube. It may be necessary to support the piston rod to reduce leverage on
the tube.
2) Reduce the flow of the gas to a slow stream, but sufficient to prevent air from
diffusing in during the experiment. Be careful with CO2 as it cools considerably
upon expansion from the cylinder.
3) The microphone is powered by a small battery which you should obtain from, and
return to, your T.A. Make sure that the polarities of the circuit, including the
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10
microphone battery, are correct.
4) Switch on the wave generator and connect its leads to the audio amplifier and
through it to the speakers and the horizontal input of the oscilloscope.
5) Switch on the oscilloscope and the power supply unit for the amplifier. Insert the
microphone battery in its holder and connect the microphone leads to drive the
vertical sweep of the oscilloscope.
6) Select an appropriate frequency (1000 Hz for CO2, air and N2. 2000 Hz for He)
and adjust the intensity of the sound using the attenuator on the signal generator. Do
not "overblast"! Details of adjustment of the oscilloscope will be provided by the
T.A. as the instruments may not be of the same type. Ideally the components should
be adjusted so that the two signals to the oscilloscope are about equal in magnitude
and a circle is seen on the screen when the signals are 90o out of phase.
7) Slowly move the piston towards the speaker until a straight line pattern is
observed on the screen. Record the position of the piston and note whether the
phase shift is 0o or 180o (0o shift occurs when the line is inclined 45o to the right and
the 180o phase shift occurs when the line is inclined 45o to the left).
8) Move the piston inward again until a next straight line pattern is obtained. Note
the position of the piston and the phase shift.
9) Continue taking as many readings as the geometry of the tube allows. Record the
temperature of the gas at the inlet. If necessary, repeat the entire procedure until
satisfactory results are obtained.
10) Repeat with the other gases provided. If time permits do the measurements at a
different frequency.
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Heat Capacity Ratio for Gases (g)
11
3.3 Calculations:
Average the spacings l/2 and using the known frequency calculate g for the
gas from equation (4.20). Compare these experimental values with the theoretical
values calculated from the Principle of Equipartition of Energy (3) and with the
accepted literature values (4). Calculate the standard deviation s for g on the
assumption that the main sources of error are due to uncertainty in l and f. The
uncertainty of the latter may be taken as 0.5% of the reading.
3.4 Discussion:
Discuss your results in terms of the Microscopic Interpretation of the Heat
Capacities of Gases.
4.
MICROSCOPIC INTERPRETATION OF THE HEAT CAPACITY OF
GASES
4.1 Degrees of freedom:
The number of degrees of freedom of the molecule is the number of
independent co-ordinates which must be specified in order to locate the molecule
and its component atoms in space. Thus for a molecule containing N atoms, 3N
co-ordinates will be required and the molecule has 3N degrees of freedom.
4.2 Classification of degrees of freedom:
Rather than consider each degree of freedom to be a position co-ordinate it is
convenient to classify the degrees of freedom as follows:
(a)
Translational degrees of freedom represent the three co- ordinates required
to specify the position of the centre of mass of the molecule. For a monatomic
gas, of course N = 1 and the total number of degrees of freedom will be 3, all
of which are translational.
(b)
Rotational degrees of freedom are possessed by polyatomic molecules since
different orientations in space are possible. For linear molecules we must
specify the orientation of the molecule with respect to rotation about two
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12
mutually perpendicular axes passing through the centre of mass of the
molecule perpendicular to the axis of the molecule.
Different orientations with respect to the molecular axis cannot be recognized and so
a linear molecule has only two rotational degrees of freedom.
For non-linear molecules we must specify the orientation with respect to three
mutually perpendicular axes and therefore a non-linear molecule possesses three
rotational degrees of freedom.
(c)
Vibrational degrees of freedom make up the remaining 3N-5 degrees of
freedom for linear molecules and 3N-6 degrees of freedom for non-linear
molecules. These specify the displacement of the atoms from their equilibrium
positions due to vibration.
4.3
Normal modes of vibrations are associated with each vibrational
degree of freedom and have characteristic vibrational frequencies.
Thus for linear and non-linear triatomic molecules there are,
respectively, four and three vibrational degrees of freedom and the
normal modes associated with them are classified as:
Fig. 6 Normal modes of vibration of a triatomic molecule.
Equipartition of Energy:
According to the kinetic theory of gases, the average kinetic energy per molecule for
an ideal gas is given by
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Heat Capacity Ratio for Gases (g)
1 / 2 m u2 =
3
kT
2
13
(21)
Since the energy of a monatomic gas is entirely kinetic energy of translation we can
write for the energy per mole
U =
3
RT
2
(22)
and thus for a monatomic gas
 
CV =
U
T
3
= R≈12.4 J mol −1
2
V
(23)
Why is this CV and not CP?
Polyatomic molecules possess internal degrees of freedom and the internal motions
will make a contribution to the energy of the gas, but to what extent? A monatomic
gas has only 3 degrees of freedom (translation) and if we assume that they are
independent of each other then they will each contribute 1/2 kT.
The principle of equipartition of energy states that each square term (e.g. 1/2 mu2x
for the kinetic energy in the x direction) in the expression for the total energy of the
molecule contributes 1/2 kT to the total energy.
Rotational motions have rotational kinetic energy 1/2 Iw2 for each axis of rotation,
where I = moment of inertia about the axis and w = angular velocity. Thus each
rotational degree of freedom contributes 1/2 kT to the total energy.
Vibrational motions affect both the kinetic and potential energy, each of which can
be expressed by a square term (1/2 mu2 and 1/2kx2 where m is the reduced mass and
k is the force constant) and thus each vibrational degree of freedom will contribute
kT to the total energy.
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Heat Capacity Ratio for Gases (g)
U (total) = U (trans) + U (rotation) + U (vibratio n)
14
(24)
Thus according to the classical equipartition principle the molar energy of a diatomic
molecule will be
U =
3
RT + RT + RT = 3.5RT
2
(25)
and
CV = 3.5 R = 29.1 J. mol -1
(26)
The value observed is actually close to 21 J. mol-1 for most diatomic gases. This is
the value to be expected in the absence of the vibrational degree of freedom. This is
a quantum mechanical effect. Thus, classically, one can gradually "turn on"
vibrational or rotational motions but quantum mechanically these motions increase
in jumps and it turns out that the energy required per jump for vibrational motions is
greater than the average energy available per molecule at room temperature. Thus
the vibrations are "inactive". The rotational energy quantum jump is much smaller
and rotational motions are "active" at room temperature but at lower temperatures
these can also become "inactive".
REFERENCES:
1.
R. D. Ford, "Introduction to Acoustics", Elsevier 1970, or A. P. Dowling, J. E.
F. Williams, "Sound and Sound Waves", Ellis Harwood 1983.
2.
D. P. Shoemaker, C. W. Garland, J. L. Steinfeld and J. W. Nibler,
Experiments in Physical Chemistry, 4th edition, pp. 74-85.
3.
P.W. Atkins, "Physical Chemistry", 5th ed.(1994), W.H. Freeman,pp.
703-708.
4.
Int. Crit. Tables, Vol. 5, pp. 80-83, McGraw Hill (1928). (Thode Lib. Ref. Q
199.N27).
5.
J.H. Noggle, "Physical Chemistry", 3rd ed., 1996, pp 95 - 93, 227 - 234.
6.
R.G. Mortimer "Physical Chemistry", Benjamin/Cummings, Redwood City,
Calif., 1993, pp 70ff.